The unexpected answer was that it was related to Apple’s now-offline iDisk service. Apple has now shutdown the MobileMe servers as part of the migration to iCloud, but if certain MobileMe-related settings are in ~/Library/Preferences/.GlobalPreferences.plist, Connect to Server also does a lookup for the now-offline iDisk servers in addition to the server you’re trying to connect to. Even though the iDisk service is now offline and unresolvable, the Mac will keep trying to resolve it until it times out.

The result from the user’s perspective?: “I can’t connect to the server.”

The fix is to open the Terminal and run the commands below to delete the iToolsMember and iToolsMemberDomain keys from ~/Library/Preferences/.GlobalPreferences.plist:

defaults delete -g iToolsMember
defaults delete -g iToolsMemberDomain

Once those commands have been run, connecting to your servers should start working again.